An analytic approach

• $\Pi = [\int_\gamma \omega_X]_{\gamma \in H_2(X, \mathbf{Z})} \in \mathbf{C}^{22}$ can be computed:
  • rigorously as a ball via deformation for projective hypersurfaces (Sertöz), with PeriodSuite: github.com/emresertoz/PeriodSuite
  • lefschetz-family (heuristic homology basis, Lairez-Pichon-Pharabod-Vanhove): github.com/ericpipha/lefschetz-family
  • Elsenhans-Jahnel implemented also an heuristic approach for degree 2
• Heuristically, via lattice reduction algorithms, we can find $\Lambda \subset H_2(X, \mathbf{Z})$.
• There is no obvious way to prove that our guesses are actually correct.

• Nonetheless, given $\Pi$ as a ball, one can compute $B \gg 0$ such that

  $$\operatorname{Pic}(\overline{X})_{\mid B} := \mathbf{Z} \langle \gamma \in \operatorname{Pic} \overline{X} \mid -\gamma^2_{\mathrm{prim}} < B \rangle \subseteq \Lambda \qquad \text{(Lairez-Sert\"{o}z).}$$